| L(s) = 1 | − 2.62·2-s + 1.48·3-s + 4.88·4-s + 0.915·5-s − 3.90·6-s − 2.34·7-s − 7.55·8-s − 0.784·9-s − 2.40·10-s − 2.86·11-s + 7.26·12-s + 4.85·13-s + 6.14·14-s + 1.36·15-s + 10.0·16-s + 2.11·17-s + 2.05·18-s − 1.70·19-s + 4.46·20-s − 3.48·21-s + 7.51·22-s + 2.87·23-s − 11.2·24-s − 4.16·25-s − 12.7·26-s − 5.63·27-s − 11.4·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.859·3-s + 2.44·4-s + 0.409·5-s − 1.59·6-s − 0.885·7-s − 2.67·8-s − 0.261·9-s − 0.759·10-s − 0.863·11-s + 2.09·12-s + 1.34·13-s + 1.64·14-s + 0.351·15-s + 2.51·16-s + 0.513·17-s + 0.485·18-s − 0.390·19-s + 0.998·20-s − 0.761·21-s + 1.60·22-s + 0.599·23-s − 2.29·24-s − 0.832·25-s − 2.49·26-s − 1.08·27-s − 2.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 - 0.915T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 - 2.11T + 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 - 2.87T + 23T^{2} \) |
| 31 | \( 1 - 4.12T + 31T^{2} \) |
| 37 | \( 1 + 9.79T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.88T + 53T^{2} \) |
| 59 | \( 1 + 2.10T + 59T^{2} \) |
| 61 | \( 1 - 1.94T + 61T^{2} \) |
| 67 | \( 1 - 2.44T + 67T^{2} \) |
| 71 | \( 1 - 6.20T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 3.33T + 79T^{2} \) |
| 83 | \( 1 + 4.26T + 83T^{2} \) |
| 89 | \( 1 - 5.49T + 89T^{2} \) |
| 97 | \( 1 - 7.50T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.191152773914112260378620766801, −7.76020069488089548088867078611, −6.86495813250071252765520209081, −6.15167147709821175657256896580, −5.53181659953213011626212470391, −3.75853931070430413344268614452, −2.96536038105291879763473920736, −2.33493508691910375694832024474, −1.29830334409475640850823536865, 0,
1.29830334409475640850823536865, 2.33493508691910375694832024474, 2.96536038105291879763473920736, 3.75853931070430413344268614452, 5.53181659953213011626212470391, 6.15167147709821175657256896580, 6.86495813250071252765520209081, 7.76020069488089548088867078611, 8.191152773914112260378620766801
